Euler´s formula in computing hyper-complex fourier transform

The Euler´s formula e^jθ = cos θ + j sin θ has been proved to be useful in simplifying the evaluation of complex and hyper-complex fourier transform by turning the multiplications of quaternion directly into matrices´s addition. In this paper we point out that while further considering the formula e^Â s^B̂ = e^{Â+B̂+1/2[Â,B̂]}, the above analytic computations can be proceeded farther, thus the numerical process is simplified. We also illustrate the relationship of this formula with the rotation in physics system, which is helpful for us to do the rotation when discussing the image filters using the quaternion operators. Additionally, we use a special algebraic summation convention Σ_m A_mB_m = Aⁱ Bⁱ to reduce the complexity of non-commutivity in calculating the quaternion Fourier transform and the discrete quaternion Fourier transform. This technique is expected to simplify further the corresponding numerical programme.